The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory

By Z. Bern, Michal Czakon, L. Dixon, David A. Kosower, Vladimir A. Smirnov. 2006

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We present an expression for the leading-color (planar) four-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4-2{epsilon} dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around {epsilon} = 0, and obtain analytic expressions for the poles from 1/{epsilon}{sup 8} through 1/{epsilon}{sup 4}. We give numerical results for the coefficients of the 1/{epsilon}{sup 3} and 1/e{sup 2} poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the 1/{epsilon}{sup 2} coefficient allows us to test a conjecture of Eden and Staudacher for the four-loop cusp (soft) anomalous dimension. We find that the conjecture is incorrect, although our numerical results suggest that a simple modification of the expression, flipping the sign of the term containing {zeta}{sub 3}{sup 2}, may yield the correct answer. Our numerical value can be used, in a scheme proposed by Kotikov, Lipatov and Velizhanin, to estimate the two constants in the strong-coupling expansion of the cusp anomalous dimension that are known from string theory. The estimate works to 2.6% and 5% accuracy, providing non-trivial evidence in support of the AdS/CFT correspondence. We also use the known constants in the strong-coupling expansion as additional input to provide approximations to the cusp anomalous dimension which should be accurate to under one percent for all values of the coupling. When the evaluations of the integrals are completed through the finite terms, it will be possible to test the iterative, exponentiated structure of the finite terms in the four-loop four-point amplitude, which was uncovered earlier at two and three loops.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/9143b4a3be0e4b921b73cf87701c194e9b33559a

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Citations in the corpus

• Hexagon functions and the three-loop remainder function (2013, background): “In particular, the amplitudes exhibit dual conformal symmetry and a duality to light-like polygonal Wilson loops [9–14].”
• Bootstrapping an NMHV amplitude through three loops (2014, background): “Many of the more powerful approaches toN = 4 supersymmetric scattering amplitudes compute the loop integrand of the theory [1–3, 5, 9, 10, 17–20].”
• Scattering amplitudes, Wilson loops and the string/gauge theory correspondence (2008, methodology): “An example is provided by the four-loop four-gluon planar amplitude [51].”
• From correlation functions to scattering amplitudes (2010, background): “At present we have no explanation why this is so, but if this new duality correlators/amplitudes is confirmed, it will provide the natural explanation of the mysterious dual conformal symmetry [13] of the loop momentum integrals in all available gluon amplitude calculations (n gluons at one loop [34”
• A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, background): “The duality with Wilson loops [3, 4, 5, 6, 7, 8, 9] and the associated dual conformal symmetry [7, 10, 11, 12, 13] of the planar theory mean that only amplitudes with six or more external legs are non-trivial.”
• The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM (2010, methodology): “The 4-point amplitude integrand at l = 2, 3, 4 loop-level were computed in [60], [61], and [62], respectively.”
• Erratum: On the structure of infrared singularities of gauge-theory amplitudes (2009, background): “The four-loop cusp anomalous dimension is known for N = 4 SYM in the planar limit [44].”
• The Zamolodchikov-Faddeev Algebra for AdS5 x S5 Superstring (2006, background): “A number of further important verifications has been made [16]-[19], 1 which supports the picture of the interpolating crossing-symmetric S-matrix from weak (gauge theory) to strong (string theory) coupling.”
• On string S-matrix, bound states and TBA (2007, background): “This solution successfully passed a number of non-trivial tests [19]-[34].”
• Scattering Amplitudes from Multivariate Polynomial Division (2012, methodology): “The first extension of the integrand reduction method beyond one-loop was proposed in [14], and it was used to reproduce the results of two-loop 5point planar and non-planar amplitudes in N = 4 SYM [15, 16].”